Welcome to Gaia! ::

The Physics and Mathematics Guild

Back to Guilds

 

Tags: physics, mathematics, science, universe 

Reply The Physics and Mathematics Guild
Theory of Quantum Computation

Quick Reply

Enter both words below, separated by a space:

Can't read the text? Click here

Submit

nonameladyofsins

PostPosted: Wed Oct 10, 2007 5:18 pm


well, there isn't one. Isn't that exciting? There is evidence that different kinds of quantum computers work in a similar manner but there is no equivalent to Turing's Turing Machine or Church's lambda calculus.

Any ideas?
PostPosted: Thu Oct 11, 2007 5:03 am


There's currently a few prototypes using up to 16 qubits. I wouldn't call them ready for the market, but that's not a problem for now. The big issue is in cryptography. You have, for instance, public key cryptography using the RSA integer factorization problem. It's based on the computational complexity of calculating p and q, where p and q are large prime numbers, and you are given pq. Since the first factor of these huge numbers is going to be tremendously large, and the amount of time it takes to check something like 7403756347956171282804679609742957314259318888923128908493623263897276503402826627689199641962511
7843995894330502127585370118968098286733173273108930900552505116877063299072396380786710086096962537934650563796359
is huge (take for instance the RSA integer factorization challenge), the problem is hard to solve on current computers, when the best algorithm so far (general number field sieve) is O(e^((1.9229+O(1))∗ln(n)^1/3∗ln(ln(n))2/3)). This still isn't polynomial time, which means it scales horribly with the size of the input.

Now with QM, one can use Shor's quantum algorithm to factor these bad boys, which is only O((log N)^3). Shor's algorithm basically finds all the solutions to the problem at once, both wrong and right answers, and the real magic is using quantum mechanical voodoo to make the wrong answers cancel themselves out instantly. This was already tested in 2001, factoring 15 into 3 and 5. With the advent of new resources for quantum computing (NIST and Yale have both built QM computer buses for data transfer across different parts of the system) we should see a huge increase in applications.
One can also use things like Grover's algo to search a large keyspace in O(n^1/2) instead of either a brute force search (O(n)) or a random-guess search (O((n+1)/2)). Where Shor can be used to attack integer factorization problems in public-key crypto, Grover's can be used to search a large keyspace such as AES in 256 bit mode, which is the encryption used for SECRET and TOP SECRET data privacy classifications in the US gov't, having a keyspace of 2^256 possibilities. It is a very exciting time indeed, and calls for a much needed paradigm change in the field of cryptography (elliptic curve crypto might work for a while, it's based on another problem, not integer factorisation, and for which there is currently no decently fast algorithm to crack).

As you say, there's little development in the way of quantum computing theory, but I'd say that's more because of (scientists: laugh now) lack of funding than anything else. The equipment to make the needed experiments is not nearly as ubiquitous as silicon was in the days of Apple's garage-borne beginnings, or IBM. The "if I touch it I break it" deal with QM is one of the issues, as is finding ways to measure the data. This is all considering it's so far lab-only, and a working commercial product may well be decades ahead. The system needs to be completely sheltered from any outside interventions. Compare this with the One Laptop Per Child initiative, where the classical system is made to withstand pretty rough conditions.

All in all, I'd guess that with more funding, more research and above all more tangible results, we'll see a spike in CS interest in developing an equivalent of turing's work on classical systems, on QM based ones. As for a normal timeframe, I'd wager 10 years before enough interest gathers momentum to make such a theory, and make such working examples.

Rayquazza


nonameladyofsins

PostPosted: Thu Oct 11, 2007 5:18 pm


why does the algorithm have an O time factor when the collapse of the wave function is supposed to be instantaneous!!!

Also, classic computer theory was there waaay before actual computers were made, all we need is quantum logic for a computer theory, and a couple of ingenous people.
PostPosted: Thu Oct 11, 2007 6:18 pm


Because Shor's is divided into a classical part, and a QM part. The part where he makes use of fun QM is the second one, where he discards the inconsistent answers. The first part is still classical, where he finds severa members of a sequence (r, x^r mod N, for 1<=r<=N), and how we find the prime factors of the beast of a number N is by using a shared property of all of these sequence members. Clasically, you'd have to compute it for every single member separatedly. Fortunately, with QM the second step of this part is easier, since you use the fact that they're in a superposition to make the wrong answers cancel eachother out, leaving only the correct ones.
This guy explains it much better, and without using any advanced math. The part where he describes the quantum fourier transform is a bit sketchy, but it works.

As for quantum logic, we already have a fre quantum logical gates and their corresponding operations (Hadamard gate for instance). But yes, we need some more interest in the field, since using probabilistic algorithms is a bit harder to describe, in CS theory, than using boolean algebra as was the case with traditional transistor-based computers.

Rayquazza

Reply
The Physics and Mathematics Guild

 
Manage Your Items
Other Stuff
Get GCash
Offers
Get Items
More Items
Where Everyone Hangs Out
Other Community Areas
Virtual Spaces
Fun Stuff
Gaia's Games
Mini-Games
Play with GCash
Play with Platinum